A subfactor is simple if it has no non-trivial normal intermediate subfactor.
A group subfactor is simple iff the group is simple. A maximal subfactor is a fortiori simple.

Question: Let $(N \subset M)$ be a subfactor, and let $$ N=K_1 \subset K_2 \subset \dots \subset K_r = M $$ be a normal chain such that each subfactor
$(K_i \subset K_{i+1})$ is simple, and $K_i \neq K_{i+1}$ for $0<i<r$.
Then any other normal chain of $(N \subset M)$ having the same properties is
equivalent to this one (i.e. the sequence of subfactors in our two
chains are the same up to isomorphisms, and a permutation of the
indices) ?

In the lattice theory framework, the subfactors $(A \subset B)$ and the isomorphisms, are replaced by intervals $[a,b]$ and projectivities (two intervals $[a, b]$ and $[c, d]$ are perspective if $b ∨ c = d$ and $b ∧ c = a$ or vice versa. Projectivity is the transitive closure of perspectivity). There is a well-known Jordan-Hölder theorem for modular lattices (also semimodular, see this paper of Grätzer-Nation).

So we would need, firstly to prove that the set of normal intermediate subfactors is a lattice and is modular, and secondly that projective intervals in such lattices give isomorphism of subfactors.

An inclusion of groups is called simple if it admits no non-trivial normal intermediate subgroups.Examples: the maximal inclusions $(H \subset G)$ are simple and $(\{ e\} \subset G)$ is simple iff $G$ is simple.

For the rest of the answer, $K$ and $L$ are normal intermediate
subgroups of the inclusion $(H \subset G)$.

Definition: $(H \subset G)$ is of class $\mathcal{C}$ if for all $K$, $L$ and $\forall k \in K$, $k.core_{KL}(K) \cap L \neq \emptyset$.
For the rest of the answer, the inclusion $(H \subset G)$ is supposed to
be of class $\mathcal{C}$.

Theorem: The Jordan-Hölder theorem is true for the class $\mathcal{C}$ group-subgroup subfactors.Proof: It's a consequence of the results above and what it's explained in the second part of the post. $\square$

Problem: Are all the inclusions $(H \subset G)$ of class $\mathcal{C}$ (see here)?Examples: If $(H_i \subset G_i)$ is a maximal inclusion, then it is obviously of class $\mathcal{C}$, and $(H_1 \times H_2 \subset G_1 \times G_2)$ is also of class $\mathcal{C}$ (see here).

The Jordan-Hölder property is true for large classes of subfactors: see class $\mathcal{C}$ and beyond, above,
but it's false in general, counter-examples are given by $(A_n \subset S_{n+1})$, see this answer:

Because there are non-equivalent inclusions of groups which are isomorphic as group-subgroup subfactors (see here), we need to add that $(R^{S_{n+1}} \subset R^{S_{n}}) \not\simeq (R^{A_{n+1}} \subset R^{A_{n}})$.

In fact, more generally we have the following properties (see M Izumi here, 23:30 and 27:50):
If $(R^{B} \subset R^{A}) \simeq (R^{D} \subset R^{C})$ then $Rep(A/B_A) \simeq Rep(C/D_C)$, with $A_B$ the normal core.
If moreover the inclusions are maximal then $(A \subset B) \sim (C \subset D)$